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###Simulating Trajectories

tmax <- 10
X1_0 <- 600
X2_0 <- 30
X3_0 <-10^5
rho <- 0.108
delta <- 0.5
eta <- 9.5*10^(-6)
lambda <- 36
N1 <- 1000
C <- 3
sigma1 <- 0.1
sigma2 <- 0.1
sigma2 <- 0.1

make_path <- function(tmax, x1_0, x2_0, x3_0, rho, delta, eta, lambda, N1, C, sigma1, sigma2, sigma3, deltat, n){
  #Initializing
  nosteps <- tmax/deltat
  x1 <- numeric(nosteps + 1)
  x2 <- numeric(nosteps + 1)
  x3 <- numeric(nosteps + 1)
  x <- matrix(data = c(x1, x2, x3), ncol = 3)
  x[1,] <- c(x1_0, x2_0, x3_0)
  sigma <- c(sigma1, sigma2, sigma3)
  
  #Simulationg the driving BM
  dW1 <- rnorm(n = nosteps, mean = 0, sd = sqrt(deltat))
  dW2 <- rnorm(n = nosteps, mean = 0, sd = sqrt(deltat))
  dW3 <- rnorm(n = nosteps, mean = 0, sd = sqrt(deltat))
  dW <- matrix(data = c(dW1, dW2, dW3), ncol = 3)
  
  #Defining drift function
  drift <- function(x){
    x1 <- x[1]
    x2 <- x[2]
    x3 <- x[3]
    y1 <- lambda - rho*x1 - eta*x1*x3
    y2 <- eta*x1*x3 - delta*x2
    y3 <- N1*delta*x2 - C*x3
    y <- c(y1, y2, y3)
    y
  }
  
  #Iterating
  for (k in (1:(nosteps))){
    x[k+1,] <- x[k,] + drift(x[k,])*deltat + sigma*x[k,]*dW[k,] + 0.5*sigma*x[k,]*sigma*(dW[k,]*dW[k,] - deltat)
  }
  
  #Creating output and adding time variable
  t <- (0:nosteps)*deltat
  out <- data.frame(x, t)
  colnames(out) <- c("x1", "x2", "x3", 't')
  out
}

subsample <- function(x){
  x <- as.matrix(x)
  
  x_fine <- matrix(data = 0, nrow = 500, ncol = 4)
  for (i in (1:500)){
    x_fine[i,] <- x[20*(i-1) + 1,]
  }
  x_fine <- as.data.frame(x_fine)
  colnames(x_fine) <- c('x1','x2','x3','t')
  
  x_coarse <- matrix(data = 0, nrow = 100, ncol = 4)
  for (i in (1:100)){
    x_coarse[i,] <- x[100*(i-1) + 1,]
  }
  x_coarse <- as.data.frame(x_coarse)
  colnames(x_coarse) <- c('x1','x2','x3','t')
  
  out <- list(x_fine, x_coarse)
  out
}

path <- make_path(tmax = 10, x1_0 = 600, x2_0 = 30, x3_0 = 10^5, rho = 0.108, delta = 0.5, eta = 9.5*10^(-6), lambda = 36, N1 = 1000, C = 3, sigma1 = 0.1, sigma2 = 0.1, sigma3 = 0.1, deltat = 0.001)
sampled_path <- subsample(path)
x_fine <- sampled_path[[1]]
x_coarse <- sampled_path[[2]]

###Plotting the trajectory

path_plot_fine <- plot_ly(x_fine, x = ~x1, y = ~x2, z = ~x3,
                          type = 'scatter3d', mode = 'lines+markers',
                          line = list(width = 6, color = ~t, colorscale = 'Viridis'),
                          marker = list(size = 3.5, color = ~t, colorscale = 'Greens', cmin = -20, cmax = 50))
path_plot_fine

path_plot_coarse <- plot_ly(x_coarse, x = ~x1, y = ~x2, z = ~x3,
                          type = 'scatter3d', mode = 'lines+markers',
                          line = list(width = 6, color = ~t, colorscale = 'Viridis'),
                          marker = list(size = 3.5, color = ~t, colorscale = 'Greens', cmin = -20, cmax = 50))
path_plot_coarse

Euler-Maruyama Estimator

For making inference, we need to have a likelihood for a ny given parameter \(\theta\). This will be the sum of the likelihoods of each step, under the EM-hypothesis. Thus, for each increment, we need to assign a likelihood of that increment, given the parameter vector. In this case, the stochastic increments to each coordinate are independent, due to the diagonal structure of \(\Sigma\). So the stochastic quantities here are Normal and Chi-square distributed. But we have a sum here, but one is a deterministic transformation of the other. But there might be more that one dW that give the same increment. In that case i suppose we have to assign both the likelihoods. Do we need to take a convolution to find the density of the sum? Apparently, we are to use a Gaussian approximation as given on the slides. The books probably cover this as well. Upon looking at the slides again, the EM-based estimator seems quite doable.

drift <- function(x){
    x1 <- x[1]
    x2 <- x[2]
    x3 <- x[3]
    y1 <- lambda - rho*x1 - eta*x1*x3
    y2 <- eta*x1*x3 - delta*x2
    y3 <- N1*delta*x2 - C*x3
    y <- c(y1, y2, y3)
    y
  }
likelihood_step <- function(theta, x, x_next){
  # need to assign subparameters of theta to the names used in the code
  y <- drift(x)
  a <- log(((2*pi*deltat)^3) * (x[1]*x[2]*x[3]*sigma1*sigma2*sigma3)^2)
  b <- ((x_next - x + y*deltat)^2)*c((x[1]*sigma1)^(-2), (x[2]*sigma2)^(-2), (x[3]*sigma3)^(-2))/deltat
  }
  a+b

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